Graphs are an obvious candidate for a topological approach. We can very naturally define simplicial complexes using the vertices and edges. More challenging is understanding how to best utilise directions within a directed graph - there are many non-equivalent generalisations. Developing and understanding TDA methods for directed graphs is important as directed graphs are common in aplications from neuroscience to social networks.
Some papers involving the topological analysis of directed graphs
Rips filtrations for quasimetric spaces and asymmetric functions with stability results
K Turner - Algebraic & Geometric Topology, 2019
Techniques in topological data analysis are suited to analysing metric spaces and symmetric functions as the standard building blocks are symmetric. Undirected networks create symmetric functions, and directed networks asymmetric functions. This paper explores 4 ways to generalise from symmetric to asymmetric functions and shows they are stable under noise.
Cliques of neurons bound into cavities provide a missing link between structure and function
MW Reimann, M Nolte, M Scolamiero, K Turner, R Perin, G Chindemi, P Dłotko, R Levi, K Hess, H Markram - Frontiers in computational neuroscience, 2017
Graph Pseudometrics from a Topological Point of View
AL Garcia-Pulido, K Hess, J Tan, K Turner, B Wang, N Yerolemou - arXiv preprint arXiv:2107.11329, 2021